domain transfer via cross-domain analogy matthew klenk and ken

Domain Transfer via Analogy 1

Running head: DOMAIN TRANSFER VIA CROSS-DOMAIN ANALOGY

Domain Transfer via Cross-Domain Analogy

Matthew Klenk and Ken Forbus

Qualitative Reasoning Group, Northwestern University

2133 Sheridan Rd, Evanston, IL 60201 USA

{m-klenk, forbus}@northwestern.edu


Abstract

Analogical learning has long been seen as a powerful way of extending the reach of one‟s

knowledge. We present the domain transfer via analogy (DTA) method for learning new

domain theories via cross-domain analogy. Our model uses analogies between pairs of textbook

example problems, or worked solutions, to create a domain mapping between a familiar and a

new domain. This mapping allows us to initialize a new domain theory. After this initialization,

another analogy is made between the domain theories themselves, providing additional

conjectures about the new domain. We present two experiments in which our model learns

rotational kinematics by an analogy with translational kinematics, and vice versa. These learning

rates outperform those from a version of the system that is incrementally given the correct

domain theory.


Domain Transfer via Cross-Domain Analogy

1 Introduction

Cognitive scientists have long argued that cross-domain analogy is an important element

in people‟s adaptability to new situations. It has been studied in the contexts of learning new

domains quickly (Gentner & Gentner 1983; Gentner 2003), producing paradigm shifts in

scientific thought (Gentner et al. 1997; Holyoak & Thagard 1989; Falkenhainer 1988), and

affecting future learning in new domains (Rand et al. 1989). While analogies are powerful, Gick

and Holyoak (1980) found that people have difficulties spontaneously producing cross-domain

analogies, but they succeed when given hints. Collins & Gentner (1987) report that successful

cross-domain analogies require a known base domain and a domain mapping. A domain

mapping consists of correspondences between the objects and relationships of the two domains.

Additional evidence of the utility of cross-domain analogies comes from textbook authors, who

routinely use them to explain concepts. The linear kinematics section of the physics textbook

used for this work (Giancoli 1991) contains eight worked through examples, or worked solutions.

These include instantiations of all four linear kinematics equations. In the rotational kinematics

section, however, there are only two worked solutions, neither of which utilizes two of the

equations necessary for completing problems from this chapter. The summary section of the

rotational motion chapter invites the learner to use analogy to fill in the details: “The dynamics

of rotation is analogous to the dynamics of linear motion” (p. 197, Giancoli 1991). Our model

seeks to capture this kind of learning.

One method students use to learn physics is by studying worked solutions. While these

step-by-step explanations are not deductive proofs, they do provide students with valuable

feedback. An insight of this work is that the structure of these explanations provides aspects of


underlying structure of the domain. Our hypothesis is that we can use this structure to learn

domain mappings between physics domains. Our model, domain transfer via analogy (DTA),

uses worked solutions and a known base domain as a starting point for learning a novel domain

theory through multiple cross-domain analogies. DTA learns a domain mapping through an

analogy between worked solutions from the known and novel domains. Our model uses this

domain mapping to initialize the new domain theory and constrain an analogy between the base

and novel domain theories. This analogy results in inferences about the new domain that are

added to the agent‟s knowledge after verification. This paper describes DTA, which uses

existing computational models of analogy and similarity based-retrieval to learn new domain

theories.

We begin by discussing the structure-mapping theory of analogy and the computational

models used in this work. Next, we describe the representations for the problems, worked

solutions and domain theories our model uses. We then outline the DTA method and illustrate it

with an example. We evaluate our model via two experiments, in which our model learns

rotational kinematics by an analogy with translational kinematics, and vice versa. We compare

these learning rates against a version of the system that is incrementally provided with the

correct domain theory. We close with a discussion of related work and implications for future

work.

2 Structure-mapping and Analogy

Before discussing the DTA method, it is necessary to describe the structure-mapping

processes of analogy and similarity-based retrieval which are used by our model. We use

Gentner‟s (1983) structure-mapping theory, which postulates that analogy and similarity are

computed via structural alignment between two representations (the base and target) to find the


maximal structurally consistent match. A structurally consistent match is based upon three

constraints: tiered-identicality, parallel connectivity, and one-to-one mapping. The tiered-

identicality constraint provides a strong preference for only allowing identical predicates to

match, but allows for rare exceptions (e.g., the minimal ascension method described below).

Parallel connectivity means that if two statements are matched then their arguments must also

match. The one-to-one mapping constraint requires that each element in the base corresponds to

at most one element target, and vice versa. To explain why some analogies are better than

others, structure-mapping uses the principle of systematicity: a preference for mappings that are

highly interconnected and contain deep chains of higher order relations.

The Structure Mapping Engine (SME) simulates the structure-mapping process of

analogical matching between a base and target (Falkenhainer et al. 1989). The output of this

process is one or more mappings. A mapping is a set of correspondences representing a

construal of what items (entities and expressions) in the base go with what items in the target.

Mappings include a structural evaluation score indicating the strength of the match, and

candidate inferences which are conjectures about the target using expressions from the base

which, while unmapped in their entirety, have subcomponents that participate in the mapping‟s

correspondences. SME operates in polynomial time, using a greedy merge algorithm (Forbus &

Oblinger 1990). If one mapping is clearly the best, it produces only that mapping, but it can

produce up to three if there are close competing matches. Pragmatic constraints can be specified

to guide the alignment process, e.g., one can state that particular entities and/or statements must

match, or must not match. We use this capability to guide abstract domain mappings with

correspondences found via the more concrete worked solution mappings.


MAC/FAC (Forbus et al. 1995) is a computational model of similarity-based retrieval.

The inputs are a set of predicate calculus facts, the probe case, and a case library. The first stage

(MAC) uses a computationally cheap, non-structural matcher to filter candidate cases from the

case library, returning up to three if they are very close. The second stage (FAC) uses SME to

compare the cases returned by MAC to the probe and returns the best candidate as determined by

the structural evaluation score of each match (or candidates, if they are very similar). Both SME

and MAC/FAC have been used as performance systems in a variety of domains and as cognitive

models to account for a number of psychological findings (Forbus 2001).

Different domains are often represented using different predicates, especially when they

are first being learned and underlying commonalities with previous knowledge have yet to be

found. In this work, we use minimal ascension (Falkenhainer 1988) to match non-identical

predicates. If two predicates are part of a larger aligned structure and share a close common

ancestor in the taxonomic hierarchy, then SME can include them in the mapping. Figure 1

demonstrates an example of two expressions that are placed in correspondence because they

have identical predicates, stepUses. The entities for the objects, events, and steps are already in

correspondence, given the rest of the mapping. In order to include these expressions in the

mapping, primaryObjectMoving would have to map to objectRotating. Minimal ascension

allows this mapping because both relationships are children of objectMoving in the

ResearchCyc1 ontology, the taxonomic hierarchy used in this work. This allows a looser form of

semantic similarity, in addition to identicality, the strongest form, to guide the matching process.

3 Representations and Problem Solving

Before we describe how cross-domain analogy enables the learning of new domain

theories, we must describe the representation of the problems, worked solutions, and domain


theories used by our model. The representations used in this work are in CycL, the predicate

calculus language of the ResearchCyc knowledge base (Matuszek et al. 2006). The

representations use the ontology of ResearchCyc, plus our own extensions. These concern QP

theory (Forbus 1984) and problem-solving strategies, and are small compared to the 30,000+

concepts and 8,000+ predicates already defined in the KB. Thus, objects, relations, and events

that appear in physics problems such as “rotor”, “car”, and “driving” are already defined in the

ontology for us, rather than being created specifically for this project. This reduces the degree of

tailorability in our experiments.

3.1 Example Problem and Worked Solution

All the problems used in this work were taken from the same physics textbook (Giancoli

1991). We represent the problems and worked solutions as cases, consisting of sets of predicate

calculus facts. The problem representations are intended to be direct translations into predicate

calculus from natural language problem statements, without any abstraction or reasoning.

Consider the problem of “How long does it take a car to travel 30m if it accelerates from rest at a

rate of 2 m/s2?” (Example problem 2-6, p. 26). This problem is represented in our system as a

case of nine facts, shown in Figure 2. The first two facts define the entities in the problem, a

transportation with land vehicle event, Acc-2-6, and an automobile, Car-2-6. The next 4 facts

describe the motion of Car-2-6 during Acc-2-6. The last 3 facts describe the question of the

problem and how these facts are grouped together in a case.

Worked solutions are represented at the level of explained examples found in textbooks.

They are neither deductive proofs nor problem-solving traces produced by our solver. Because

the textbook has only a limited number of worked solutions for rotational kinematics, in


instances where there are no worked solutions, we created our own from problems at the end of

the chapter. Below is an English rendering of the worked solution for this example problem:

1. Categorize the problem as a constant acceleration linear mechanics problem

2. Instantiate the distance by velocity time equation (d = vit + .5at2)

3. Because the car is stationary at the start of the event infer that its velocity is zero (vi = 0

m/s)

4. Solve the equation for t (t = 5.8s)

The entire worked solution consists of 38 facts. Figure 3 shows the predicate calculus

representation for the third worked solution step. The first four facts indicate the type of solution

step and its sequential position in the worked solution. The last two facts state that the step uses

the fact that Car-2-6 is stationary at the beginning of Acc-2-6 to infer that its speed at that

point is 0m/s. Later, we show how DTA uses analogies between worked solutions from different

domains to learn a domain mapping.

3.2 Domain Theories for Problem Solving

Our domain theories consist of encapsulated histories (Forbus 1984) representing

equations. Encapsulated histories, unlike model fragments, permit constraints to be placed on

the duration of events and time intervals. This is necessary for representing equations, such as

the velocity/time law required to solve the problem in Figure 2 which involves events as well as

their durations. During problem solving, our system instantiates applicable encapsulated

histories to determine which equations are available.

Figure 4 shows the definition for the encapsulated history representing the equation

vf=vi+at, velocity as a function of time. There are two participants, theObject and theEvent,

which must satisfy their type constraints, the abstractions PointMass and


Constant1DAccelerationEvent, respectively. Furthermore, the conditions of the encapsulated

history must be satisfied in order to instantiate it and conclude its consequences. In this case, it

is necessary that theObject be the object moving in theEvent. The compound form shown in

Figure 4 is automatically translated into a set of predicate calculus facts for use in our system.

This knowledge is necessary for our physics problem-solver to successfully answer questions.

In addition to understanding physics phenomena and equations, solving physics problems

requires a number of modeling decisions. For example, in a scenario of dropping a ball off a

building, one must view the ball as a point mass, the falling event as a constant acceleration

event, and that the acceleration of the ball as equal to Earth‟s gravity. Our system currently uses

hand coded rules to make these decisions. While this is sufficient for the goals of this work, our

future plans involve integrating more robust modeling decision methods.

The objective of domain transfer via analogy (DTA) is learning domain theories that are

represented with schema-like knowledge. In this work, DTA learns the encapsulated histories of

a new domain via cross-domain analogy, in terms of participants, conditions and consequences.

We evaluate the domain theories learned through DTA on new problems. The physics problems

in this work all ask for the values of specific quantities. Our system solves for quantities in three

ways. First, the quantity may be given directly in the problem. Second, a modeling decision

could indicate the appropriate value for the quantity (e.g. the object is in projectile motion on

Earth, and air resistance is ignored; Therefore, our system assumes the acceleration on the object

to be 10 m/s2). Third, our system is able to instantiate an encapsulated history with a

consequence of an equation mentioning the desired quantity. In this case, our system recursively

solves for the other parameters in the equation before solving the equation for the sought after

quantity using algebra routines inspired by the system described in Forbus & De Kleer (1993).


4 Domain Transfer via Analogy

Domain transfer via analogy (DTA) learns a new (target) domain theory using multiple

cross-domain analogies. In this work, DTA transfers encapsulated histories from a known base

domain to the target domain theory. DTA is invoked after failure to solve a problem in the new

domain. Our model is provided the worked solution for that problem. Therefore, the inputs to

the DTA algorithm are a known base domain and a worked solution from the new domain. The

known base domain consists of a set of worked solutions and a domain theory, which is

represented in this work as a set of encapsulated histories. DTA uses this worked solution to

create conjectures about knowledge in the new domain, via the algorithm outlined in Figure 5.

Recall that Cognitive Science research has identified the importance of domain mappings, i.e.,

the mapping of types and relations between two domains, for successful cross-domain analogies.

Therefore the first step in our algorithm is to learn the domain mapping. Next DTA initializes

the target domain theory, extends this new domain theory using this mapping, and verifies the

learned knowledge. We describe each of these steps in detail below.

4.1 Step 1: Learn the Domain Mapping

Because different domains are represented with different predicates and conceptual types,

a domain mapping is essential to successful cross-domain analogies. DTA learns the domain

mapping through an analogical mapping between worked solutions from the two domains.

Using the worked solution for the failed problem as a probe, MAC/FAC selects an analogous

worked solution from a case library containing worked solutions from the known domain. Next,

SME creates an analogy between the retrieved worked solution and the worked solution to the

failed problem. The output of this process is up to three mappings, each containing a set of

correspondences which are used to form the domain mapping. Because SME can produce


multiple mappings, our model sorts the mappings by their structural evaluation scores.

Beginning with the best mapping, we add all of the correspondences in which the base element is

a non-numeric entity (e.g., types, quantities, and relations) mentioned in the base domain theory.

Then, our model iterates through the rest of the mappings adding correspondences to the domain

mapping that do not violate the one-to-one constraint. For example, if the best mapping had a

correspondence between PointMass and RigidObject and the second mapping had a

correspondence between PointMass and Ball, the domain mapping would only include the

mapping between PointMass and RigidObject. The reason for combining multiple mappings

is that each mapping may only cover some aspects of the worked solutions.

4.2 Step 2: Initialize the Target Domain Theory

Recall that when the system attempts the first problem in a new domain, its theory for

that domain is empty. That is, it does not have any encapsulated histories to understand the

phenomena of the new domain. DTA uses the domain mapping to initialize the new domain

theory. For each encapsulated history from the base domain theory mentioned in the domain

mapping, our model attempts to create an encapsulated history in the target domain. Before

transferring the encapsulated history, our model verifies that all of the quantities and types

mentioned in the encapsulated history appear in the domain mapping. If they do not, this

encapsulated history is not transferred to the target domain theory. If they are, then using the

domain mapping, our model proceeds by substituting concepts in the base encapsulated history

with the appropriate concepts from the target domain. The resulting encapsulated history is then

added to the target domain theory.


4.3 Step 3: Extend the Target Domain Theory

After initializing the target domain theory, DTA extends it through a second cross-

domain analogy. This time the analogy is between the base and target domain theories

themselves. The domain theories consist of the facts representing the encapsulated histories.

Our model constrains this match with the domain mapping to ensure the consistency of the target

domain theory. One result of SME is a set of candidate inferences, i.e., conjectures about the

target, using expressions from the base which are partially mapped. Because the base domain

theory is made up encapsulated histories, these candidate inferences describe corresponding

encapsulated histories in the target domain theory. Before adding these encapsulated histories to

the target domain theory, it is necessary to resolve the skolem entities mentioned in the candidate

inferences. SME creates skolem entities for entities appearing in the base which have no

correspondence in the mapping. If the skolem represents a number from the base, then that

number is used in the target. Otherwise, our model creates a new entity for each skolem to be

assumed in the target domain theory. After this process, the candidate inferences representing

new encapsulated histories are stored into the target domain theory.

4.4 Step 4: Verify the Learned Knowledge

While powerful, cross-domain analogies are risky and frequently contain invalid

inferences. Therefore, DTA verifies the newly proposed encapsulated histories by retrying the

problem whose failure began the entire process. If this problem is solved correctly, our system

assumes that the newly acquired domain theory is correct. Otherwise, our model forgets both the

new domain theory and the domain mapping, and attempts the entire process one more time,

removing the analogue worked solution from the case library so that the next-best worked

solution is retrieved.


Currently, we only let the model try two different analogous worked solutions to learn an

acceptable target domain theory. In the future, this parameter could be under control of the

system employing DTA. Next, we describe an example of our model‟s operation.

4.5 Example

To better understand how DTA uses multiple cross-domain analogies to transfer domain

theories, we describe an example of how it learns rotational kinematics through an analogy to

linear kinematics. The system begins with a linear kinematics domain theory and worked

solutions. Because the system has no encapsulated histories of rotational kinematics, it fails to

solve the following problem, “Assuming constant angular acceleration, through how many turns

does a centrifuge rotor make when accelerating from rest to 20,000 rpm in 5 min?” Because the

system failed, we invoke DTA by providing the worked solution to this problem.

In order to create a domain mapping between linear and rotational kinematics, DTA

creates an analogy between a linear kinematics worked solution and the provided rotational

kinematics worked solution. Using the provided worked solution as a probe, the model uses

MAC/FAC to retrieve an analogue from its case library of linear kinematics worked solutions.

In this case, the analogous worked solution retrieved is for the problem discussed previously,

“How long does it take a car to travel 30m if it accelerates from rest at a rate of 2m/s2?” Our

model uses an analogy between these two worked solutions to produce the domain mapping

necessary for cross-domain analogy. In this case, the mathematical relationships are isomorphic,

d = vit + .5at2 and θ=ωit + .5αt

2, which places the quantities between the domains into

correspondence. It should be noted that SME handles partial matches, allowing correspondences

to be created even when the mathematical relationships in the problems being compared are not

completely isomorphic. The minimal ascension example from Figure 1 placing


primaryObjectMoving in correspondence with objectRotating is also part of this mapping.

Next, the mapping‟s correspondences are extracted to create the domain mapping, a subset of

which appears in Table 1.

After learning the domain mapping, the next step is to initialize the target domain theory.

This is done by searching the domain mapping for encapsulated histories from the base domain.

In this example, DistanceByVelocityTime-1DConstantAcceleration is found. In order to

transfer this encapsulated history to the rotational kinematics domain theory, all of its participant

types and quantities must participate in the domain mapping. This encapsulated history contains

two participant types, PointMass and ConstantLinear-AccelerationEvent and four

quantities, Acceleration, Speed, Time-Quantity and DistanceTravelled. All of these

appear in the domain mapping allowing DTA to transfer this encapsulated history to the target

domain. For each fact in the base domain theory mentioning DistanceByVelocityTime-

1DConstantAcceleration, we substitute all subexpressions based upon the domain mapping.

This results in a new encapsulated history, DistanceTime-Rotational, which represent the

rotational kinematics equation, θ=ωit + .5αt2. Our model initializes the target domain theory by

adding this new encapsulated history.

Next, DTA extends the new domain theory with a cross-domain analogy between the

base and target domain theories themselves. To maintain consistency, this analogy is

constrained with the domain mapping acting as required correspondence constraints. The 41

facts describing the linear mechanics encapsulated histories make up the base, and the 6 facts of

the newly initialized rotational mechanics domain theory are the target. As expected, the sole

target encapsulated history maps to the corresponding linear mechanics encapsulated history.

This mapping includes the quantities, conditions and types of these encapsulated histories.


These correspondences result in candidate inferences involving the facts of the other

encapsulated histories from the base. DTA uses these candidate inferences to infer rotational

kinematics encapsulated histories. For example, the candidate inference shown in Figure 6

suggests that there is an encapsulated history in the target analogous to the VelocityByTime-

1DConstantAcceleration linear mechanics encapsulated history. This candidate inference

states that the suggested encapsulated history has the operating condition of its object rotating

during its event. The AnalogySkolemFn expression indicates that there was no corresponding

entity in the target. Therefore, to extend the target domain theory, DTA creates entities for all

the analogy skolems, i.e. turning (AnalogySkolemFn VelocityByTime-

1DConstantAcceleration) into EHType-1523, and assumes these facts into the rotational

mechanics domain theory. During this step, DTA assumes three new encapsulated histories into

the rotational kinematics domain theory.

Finally, we need to verify that the learned knowledge is accurate. This is done by

attempting to solve the original problem again. Since the worked solution contains the answer,

we can compare our computed answer against it. If they match, then we assume that the learned

knowledge is correct. If the system gets the problem wrong, it takes two steps. First, it erases

the domain mapping and the inferred encapsulated histories in the rotational mechanics domain

theory. Second, it repeats the entire process one more time, with the next-best retrieval from the

case library. Abandoning, rather than debugging, a mapping may be unrealistic. We plan to

explore diagnosis and repair of faults in our domain theories and domain mappings in future

work, as explained in Section 7. In this case, the system used the learned encapsulated histories

to get the problem correct.


Recall at the beginning the system had zero encapsulated histories in its rotational

kinematics domain theory. While the rotational kinematics worked solution contained an

example of one encapsulated history, after employing DTA, the system learned four rotational

kinematics encapsulated histories. Now, these are available for solving future problem-solving

episodes. In the next section, we evaluate DTA by giving our system a sequence of problems

from a new domain and measuring its ability to answer questions correctly.

5 Evaluation

To examine how well DTA works, we need a baseline. Our baseline spoon-fed system

consists of the exact same problem-solver. Instead of providing the spoon-fed system a worked

solution after failing a problem, we provide it with the general encapsulated histories needed to

solve that specific problem. We performed two experiments, one to evaluate learning of

rotational kinematics by an analogy with linear kinematics, and the other to evaluate learning of

linear kinematics based upon rotational kinematics. Both systems begin with the necessary rules

for problem-solving strategies and modeling decisions. The systems are then tested on a series

of problems from the target domain.

The problems for both domains are listed in Figure 7. For these problems, we created

representations and worked solutions in the manner described in section 3. For linear

kinematics, the representations for the problems and worked solutions averaged 15.4 and 52 facts

respectively. These representations included 38 different types and 52 unique relations. For

rotational kinematics, the problem and worked solution representations averaged 9.8 and 46.2

facts respectively. These representations included 21 types and 33 relations. 9 types and 26

relations appear in problems and worked solutions from both domains.


This evaluation seeks to answer the following questions. First, can DTA transfer the

encapsulated histories to solve problems in new domains? Second, can DTA learn better than a

baseline system which is incrementally given the correct domain theory? Third, how important

is the verification of the learned domain theory and the ability retrieval additional analogues?

5.1 Experiment 1

In this experiment, we use linear kinematics as the base domain and rotational kinematics

as the target domain. To address our research questions, we generated learning curves by

aggregating performance across the 120 trials representing every possible ordering of the five

rotational kinematics problems. Because rotational kinematics is the target domain, both systems

begin the trial without any rotational kinematics encapsulated histories. Therefore, we expect

neither system to solve the first problem on any of the trials. During each trial, when the DTA

system fails to solve a problem, the DTA method is invoked with the worked solution to that

problem. After each problem in the baseline condition, the system is given the necessary

encapsulated histories to solve the problem. These encapsulated histories, either learned by

cross-domain analogy in the DTA system or provided to the baseline system, allow the system to

answer future rotational kinematics questions. At the end of each, the system‟s knowledge was

reset.

Figure 8 compares the rotational kinematics learning rates for the DTA and baseline

conditions. The DTA system exhibited perfect transfer. That is, after studying just one worked

solution, the analogy system was able score 100% on the rest of the problems. Because the DTA

system performed perfectly, there were only 120 transfer attempts, all of which succeeded. Of

these, 72 attempts (60%) required retrieving a second worked solution to generate a successful

domain mapping. This highlights the importance of verifying the knowledge learned from the


cross-domain analogy. The performance in the baseline condition was markedly worse than

DTA. After one problem, the baseline system was only able to solve the next problem 45

percent of the time. Also, the baseline system‟s ceiling was at 80 percent. This was due to the

fact it was unable to solve rotational kinematics problem „b‟ from Figure 7 regardless of what

problems it had already seen, because none of the other problems use the same equation. DTA

overcomes this through the analogy between the domain theories themselves. This allows DTA

to infer equations not mentioned explicitly in the worked solution from the target domain.

5.2 Experiment 2

This experiment followed the same form as the previous experiment but with the opposite

base and target domains. Here, we use rotational kinematics as the base domain and linear

kinematics as the target domain. Once again, we are interested in comparing the learning rates

between our baseline system and our model of domain transfer via cross-domain analogy.

Once again, the DTA system outperformed the baseline system. Figure 9 graphs the

learning curves of the two conditions. After the first problem, the DTA system got the next

problem correct 60% of the time, compared with the 40% performance of the baseline. After

seeing two worked solutions, the analogy system scored 90% on the third problem where the

baseline system scored 80%. On problems 4 and 5, both systems performed at a ceiling of

100%. The baseline condition was able to achieve a ceiling of 100% as every equation required

was used by at least two problems. In the analogy condition, DTA was unable to transfer the

correct domain theory for two linear kinematics problems. This led to 180 transfer attempts, 120

after the first problem of the trial, 48 after failures on the second problem and 12 after the third

problem. Out of the 180 attempts, 120 (66%) were successful. Of the successful attempts, none

of them required using an additional retrieval.


5.3 Discussion

In both experiments, domain transfer via analogy outperformed the spoon-fed baseline

system. DTA learned faster and achieved a ceiling that was the same or higher than the baseline.

An analysis of the few linear kinematics transfer failures indicates that increasing the complexity

of sub-event structures and time intervals increases the difficulty in generating an appropriate

domain mapping. Linear kinematics problems „b‟ and „d‟ both contained such structures. One

requirement for successful transfer in these scenarios is that an objectRotating statement in the

rotational kinematics worked solution must correspond with a primaryObjectMoving statement

in the linear kinematics worked solution. In order for this to occur, the entities which make up

these expressions must already be in alignment. Given the structure of linear mechanics

problems „b‟ and „d‟, the events listed in the step uses statements may differ from the events and

time intervals referenced in the quantities and equations. Therefore, one aspect of our future

work is to incorporate rerepresentation strategies (Yan et al. 2003) to bring these worked

solutions into better alignment with the analogous rotational kinematics worked solutions. The

added complexity of the linear kinematics problems also slowed the baseline learning system.

Another interesting phenomena illustrated by these experiments is the difference in the

utility of retrieving an additional worked solution if the first one fails. In experiment 1, this part

of the algorithm was critical to the analogy system‟s perfect transfer. On the other hand, in

experiment 2, the additional retrievals never produced an adequate domain mapping. This

supports our intuition that the decision to retrieve additional analogues should be controlled by

the system based upon its goals.


6 Related Work

There are three main branches of related work: AI systems of analogy and case based

reasoning, cognitive science models of cross-domain analogy, and integrated reasoning systems

working towards human-level capabilities. The majority of analogical and case based systems

focus on using previous experiences to guide future problem-solving (Kolodner 1993). This

work has occurred in a variety of domains including transportation (Veloso & Carbonell 1993),

inductive theorem proving (Melis & Whittles 1999) and thermodynamics problem-solving

(Ouyang & Forbus 2006). These systems use examples to provide search control knowledge in

order to solve future problems faster. The most striking difference between DTA and these

systems is that we focus on learning new domain concepts as opposed to search control

knowledge. That is, without the analogical transfer, our system would not be able to answer any

questions in the target domain, and each analogical transfer allows our system to solve problems

it otherwise would not have been able to solve.

Focusing on learning from examples, the Cascade system models learner behavior in the

Newtonian physics problem-solving (VanLehn 1999). Cascade uses analogy in two ways. First,

as in the above systems, analogy is used to learn search control knowledge. Second, while

overly general rules are the primary method for acquiring new physics knowledge, Cascade is

able to use within-domain analogy during example studying to form new rules as a last resort.

We differ by focusing on learning by analogy and using multiple cross-domain analogies to

construct a new domain theory. Cascade has been used to model two psychological effects

during physics learning: the self-explanation effect (VanLehn et al. 1992) and learning from

faded examples (Jones & Fleischman 2001). Instead of focusing on matching individual human

data, we seek to understand how knowledge of related domains can accelerate learning in new


ones. Building upon existing cognitive models of analogical processing, DTA represents an

important step in this direction.

Cross-domain analogy research has focused on this problem of learning abstract

knowledge. The closest project to our approach is Falkenhainer‟s (1988) PHINEAS. PHINEAS

used comparisons of (simulated) behaviors to create an initial cross-domain mapping. The

mapping results in inferences that were used to create a partial theory for the new domain. Our

model differs from PHINEAS in several significant ways: (1) We use analogies between

problem explanations to drive the process, (2) We are learning quantitative, rather than

qualitative, domain theories, which require very different verification work, and (3) We are using

a more psychologically plausible retrieval mechanism, MAC/FAC. Holyoak and Thagard's

(1989) PI model used a pragmatic theory of analogy to model solving a variation of the radiation

problem through schema induction. PI only used analogy during problem-solving, and its

retrieval model was never extensively tested. On the other hand, our model makes analogies

between both example problems as well as analogies between domains themselves. The domain

analogies allow DTA to transfer domain knowledge not explicitly referenced in the particular

worked solutions used in creating the domain mapping. Moreover, DTA tests its learned

knowledge, and uses it to solve new problems from the target domain, whereas PI did neither.

A number of recent projects working towards human-level AI have built architectures

emphasizing the importance of integrating analogy with other forms of reasoning. ICARUS

employs a representational mapping mechanism to transfer plan goal decompositions between

different 2d grid games (Shapiro et al. 2008). Kühnberger et al.‟s I-Cog (2008) explores the

trade-offs between analogy and two other reasoning modules, all of which operate over noisy

data, using the Heuristic-Driven Theory Projection (Gust et al. 2006) model of analogy.


Schwering et al. (2008) identifies the importance of combining analogy with deductive and

inductive techniques for achieving human level reasoning. Kokinov‟s AMBR (2003) explores

the role of analogy in various aspects of memory, such as retrieval and distortion effects. We

agree that analogy is integral to the robustness of human reasoning. Our cognitive architecture,

Companions cognitive systems (Forbus et al. 2008), emphasizes that analogy is a common

operation as opposed to an exotic event undertaken rarely. For example, in DTA, each attempt at

transfer requires an analogy between worked solutions as well as between domain theories

themselves. We plan to integrate DTA into our Companions-based learning system (Klenk &

Forbus 2007), which utilizes within-domain analogies to formulate the models necessary to solve

AP Physics problems. By focusing on human-level tasks, we believe we will learn about the

constraints on the analogical processes themselves in addition to learning how to utilize them in

building AI systems (Forbus 2001).

7 Conclusions and Future Work

We have shown that a domain theory for solving physics problems can be learned via

cross-domain analogies, using our DTA method for cross-domain transfer. DTA is very general;

it should work with any domain where reasoning involves schema-like domain theories and

where analogous worked solutions are available. Furthermore, our experiments demonstrate that

such analogical learning can be very efficient. In fact, when the two domains are sufficiently

similar, DTA can outperform a baseline system that is incrementally given the correct domain

theory. The process of constructing domain mappings by exploiting structural similarities in

worked solutions, and using that mapping to import theories from one domain to another, is, we

believe, a general and powerful process.


There are several directions we intend to pursue next. First, we have only tested our

model with learning encapsulated histories, so we want to extend it to handle other types of

domain knowledge, such as model fragments and modeling knowledge. Currently, we are

developing a method for learning modeling decisions via generalization and incorporating it into

our domain transfer system (Klenk et al. 2008). Second, while DTA has an explicit verification

step, this does not guarantee that all the learned knowledge is correct. Therefore, we plan to

implement model-based diagnostic strategies to debug our analogically-derived domain theories,

similar to the strategies used by de Koning et al. (2000) to diagnose misconceptions in student

models. Integrating both generalization techniques for within-domain learning and diagnostic

techniques for cross-domain learning requires additional commitments as to when and how these

processes are invoked by a larger system. As indicated above, we plan to use the Companions

cognitive architecture to extend DTA, putting the learning process under control of plans in the

Executive. This will place decisions like whether to debug or abandon a problematic cross-

domain mapping under system control, and thereby extend the range of phenomena the model

can tackle.

Finally, we plan to explore a broader range of domain pairs, including domains which are

quite distant such as those found in system dynamics (Olsen 1966). While linear and rotational

mechanics are quite similar, the analogy between mechanics and electricity, two superficially

different domains, is quite systematic. This will explore the extent that similarities in worked

solution structure can learn an adequate domain mapping. Finally, an analogy between

mechanics and heat flow will stress that only certain aspects of the domain should transfer, as

there are important non-analogous aspects between these domains.2


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Author Note

Matthew Klenk and Ken Forbus, Department of Electrical Engineering and Computer

Science, Northwestern University.

This research was supported by the Cognitive Science Program of the Office of Naval

Research. The authors would like to thank the participants of the AnICA 2007 workshop for the

helpful feedback on the ideas presented here. Also, we thank Thomas Hinrichs, Kate Lockwood,

and Scott Friedman for their comments on this work and help revising this document.

Correspondence concerning this article should be addressed to Matthew Klenk, 2133

Sheridan Rd., Evanston, IL 60201. Email: [email protected]


Footnotes

1 http://research.cyc.com/

2 While the spring constants in mechanics are analogous to the inductance constants in

electricity, there is no analog for inductance in thermal dynamics.


Table 1

Domain mapping

Base Item Target Item PointMass RigidObject

ConstantLinear-AccelerationEvent ConstantRotational-AccelerationEvent

primaryObjectMoving objectRotating

Acceleration AngularAcceleration

Speed RateOfRotation

DistanceTravelled AngularDistTravelled

Time-Quantity Time-Quantity

DistanceByVelocityTime-

1DConstantAcceleration

DistanceTime-Rotational


Figure Captions

Figure 1. Minimal ascension aligns primaryObjectMoving to objectRotating

Figure 2. Example problem 2-6 representation

Figure 3. Representation for step 3, inferring that the car‟s velocity is 0m/s

Figure 4. Definition for the velocity by time encapsulated history

Figure 5. The DTA method for domain learning via cross-domain analogy

Figure 6. Candidate inference suggesting a condition of an encapsulated history type

Figure 7. Evaluation materials

Figure 8. Rotational mechanics learning curves

Figure 9. Linear mechanics learning curves


Figure 1

Base Expression: (stepUses Gia-2-6-WS-Step-2

(primaryObjectMoving Acc-2-6 Car-2-6))

Target Expression: (stepUses Gia-8-5-WS-Step-2 (objectRotating Acc-8-5 Rotor-8-5))


Figure 2

(isa Car-2-6 Automobile)

(isa Acc-2-6 TransportWithMotorizedLandVehicle)

(objectStationary (StartFn Acc-2-6) Car-2-6)

(primaryObjectMoving Acc-2-6 Car-2-6)

(valueOf ((QPQuantityFn DistanceTravelled) Car-2-6 Acc-2-6)

(Meter 30))

(valueOf (MeasurementAtFn ((QPQuantityFn Acceleration) Car-2-6) Acc-2-6)

(MetersPerSecondPerSecond 2))

(isa Gia-Query-2-6 PhysicsQuery)

(hypotheticalMicrotheoryOfTest Gia-Query-2-6 Gia-2-6)

(querySentenceOfQuery Gia-Query-2-6

(valueOf ((QPQuantityFn Time-Quantity) Acc-2-6) Duration-2-6)))


Figure 3

(isa Gia-2-6-WS-Step-3 WorkedSolutionStep)

(hasSolutionSteps Gia-2-6-WorkedSolution Gia-2-6-WS-Step-3)

(priorSolutionStep Gia-2-6-WS-Step-3 Gia-2-6-WS-Step-2)

(stepOperationType Gia-2-6-WS-Step-3

DeterminingSpecificScalarOrVectorValuesFromContext)

(stepUses Gia-2-6-WS-Step-3

(objectStationary (StartFn Acc-2-6) Car-2-6))

(stepResult Gia-2-6-WS-Step-3

(valueOf

(MeasurementAtFn ((QPQuantityFn Speed) Car-2-6) (StartFn Acc-2-6))

(MetersPerSecond 0)))


Figure 4

(def-encapsulated-history

VelocityByTime-1DConstantAcceleration

:participants

((theObject :type PointMass)

(theEvent :type Constant1DAccelerationEvent))

:conditions

((primaryObjectMoving theEvent theObject))

:consequences

((equationFor VelocityByTime

(mathEquals

(AtFn (Speed theObject) (EndFn theEvent))

(PlusFn (AtFn (Speed theObject) (StartFn theEvent))

(TimesFn

(AtFn (Acceleration theObject) theEvent)

(Time-Quantity theEvent)))))))


Figure 5

Domain Transfer via Analogy (DTA) method:

Given: Case library of base domain worked solutions, worked solution from target domain, base domain encapsulated histories

1. Learn the domain mapping

o Retrieve analogue worked solution from case library using MAC/FAC, with target worked solution as the probe

o Use SME to create an analogical mappings between the analogue worked solution and the target worked solution

o Create domain mapping by selecting correspondences from the mappings in which the base element appears in

the base encapsulated histories

2. Initialize target domain theory using the domain mapping

o For each base encapsulated history mentioned in the domain mapping, attempt to create a corresponding

encapsulated history in the target domain

3. Extend target domain theory

o Use SME to create a match between the base and the target encapsulated histories constrained by the domain

mapping

o Transfer domain theory contents using the candidate inferences

4. Verify learned domain theory by attempting the failed problem again

o If failure, go once more to step 1. Otherwise, accept new target domain knowledge as correct


Figure 6

(qpConditionOfType

(AnalogySkolemFn VelocityByTime-1DConstantAcceleration)

(objectRotating :theEvent :theObject))


Figure 7

Linear Kinematics Rotational Kinematics

a) How long does it take a car to travel 30m if

it accelerates from rest at a rate of 2m/s2?

b) We consider the stopping distances from a

car, which are important for traffic safety

and traffic design. The problem is best deal

with in two parts: (1) the time between the

decision to apply the brakes and their actual

application (the "reaction time"), during

which we assume a=0; and (2) the actual

braking period when the vehicle

decelerates. Assuming the starting velocity

is 28m/s, the acceleration is -6.0m/s2, and a

reaction time of .5s. What is the stopping

distance?

c) A baseball pitcher throws a fastball with a

speed of 44m/s. It has been observed that

pitchers accelerate the ball through a

distance of 3.5m. What is the average

acceleration during the throwing motion?

d) Suppose a ball is dropped from a 70m

tower how far will it have fallen after 3

seconds?

e) A jetliner must reach a speed of 80m/s for

takeoff. The runway is 1500m long. What

is the constant acceleration required?

a) Through how many turns does a centrifuge

rotor make when accelerating from rest to

20,000 rpm in 5 min? Assume constant

angular acceleration

b) A phonograph turntable reaches its rated

speed of 33 rpm after making 2.5

revolutions, what is its angular

acceleration?

c) Through how many turns does a centrifuge

rotor make when accelerating from rest to

10,000 rpm in 270 seconds? Assume

constant angular acceleration

d) An automobile engine slows down from

3600 rpm to 1000 rpm in 5 seconds, how

many radians does the engine turn in this

time?

e) A centrifuge rotor is accelerated from rest

to 20,000 rpm in 5 min, what is the

averaged angular acceleration?


Figure 8

0

25

50

75

100

1 2 3 4 5

Pe

rce

nt

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rre

ct

Problem Number

DTA

Baseline


Figure 9

0

25

50

75

100

1 2 3 4 5

Pe

rce

nt

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Problem Number

DTA

Baseline

domain transfer via cross-domain analogy matthew klenk and ken

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